Group #15673541 — subgroups

Subgroups — #15673541

Order 2944.

196 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 85 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13092263	C₂	1	1	yes	maximal, characteristic
2	#13092172	C₂	1	1	yes	maximal, characteristic
3	#13092171	C₂	1	2	yes	maximal
4	#13092171	C₂	1	2	yes	maximal
5	C₂₃×#336	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₈×C₂₃	C₂²	1	2	yes	—
7	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
9	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
10	C₂²×C₂₃⋊C₈	C₂²	1	2	yes	—
11	C₂₃⋊#95	C₄	1	2	yes	—
12	C₂₃⋊#95	C₄	1	2	yes	—
13	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
14	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
15	C₂²×Dic₂₃	C₂×C₄	1	1	yes	characteristic
16	C₂²×Dic₂₃	C₂×C₄	1	1	yes	characteristic
17	C₂²×Dic₂₃	C₂×C₄	1	2	yes	—
18	C₂×C₈×C₂₃	—	2	2	no	—
19	C₂×C₈×C₂₃	—	2	2	no	—
20	C₂×C₈×C₂₃	—	2	2	no	—
21	C₄²×C₂₃	—	2	2	no	—
22	C₄×Dic₂₃	—	2	2	no	—
23	C₄×Dic₂₃	—	2	1	no	—
24	C₄×Dic₂₃	—	2	1	no	—
25	C₂×C₂₃⋊C₈	—	2	2	no	—
26	C₂×C₂₃⋊C₈	—	2	2	no	—
27	C₂×C₂₃⋊C₈	—	2	2	no	—
28	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
29	C₂×C₄×C₂₃	—	2	2	no	—
30	C₂×C₄×C₂₃	—	2	2	no	—
31	C₂×C₄×C₂₃	—	2	2	no	—
32	C₂×C₄×C₂₃	#41	1	2	yes	—
33	C₂×C₄×C₂₃	#41	1	2	yes	—
34	C₂×C₄×C₂₃	#41	1	1	yes	characteristic
35	C₂×C₄×C₂₃	#41	1	1	yes	characteristic
36	C₂×Dic₂₃	—	2	1	no	—
37	C₂×Dic₂₃	—	2	1	no	—
38	C₂×Dic₂₃	—	2	2	no	—
39	C₂×Dic₂₃	—	2	2	no	—
40	C₂×Dic₂₃	—	2	2	no	—
41	C₂×Dic₂₃	—	2	2	no	—
42	C₂×Dic₂₃	—	2	2	no	—
43	C₈×C₂₃	—	4	2	no	—
44	C₂₃⋊C₈	—	4	2	no	—
45	#1840	—	23	1	no	maximal, Sylow
46	C₂²×C₂₃	#117	1	1	yes	characteristic
47	C₂²×C₂₃	#117	1	1	yes	characteristic
48	C₂²×C₂₃	#117	1	1	yes	characteristic
49	C₂²×C₂₃	#131	1	2	yes	—
50	C₂²×C₂₃	#131	1	2	yes	—
51	C₄×C₂₃	—	4	2	no	—
52	C₄×C₂₃	#126	1	2	yes	—
53	C₄×C₂₃	#126	1	2	yes	—
54	Dic₂₃	—	4	1	no	—
55	Dic₂₃	—	4	2	no	—
56	Dic₂₃	—	4	1	no	—
57	#376	—	23	1	no	—
58	#336	—	23	2	no	—
59	#336	—	23	2	no	—
60	#336	—	23	1	no	—
61	#336	D₂₃	1	1	yes	characteristic
62	C₂×C₂₃	#383	1	1	yes	characteristic
63	C₂×C₂₃	#433	1	2	yes	—
64	C₂×C₂₃	#433	1	1	yes	characteristic
65	C₂×C₂₃	#433	1	2	yes	—
66	C₂×C₂₃	#433	1	1	yes	characteristic
67	C₂²×C₈	—	23	2	no	—
68	C₂²×C₈	C₂×D₂₃	1	2	yes	—
69	C₂×C₄²	—	23	1	no	—
70	C₂×C₄²	—	23	1	no	—
71	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
72	#95	—	23	2	no	—
73	#95	—	23	2	no	—
74	C₂₃	#1840	1	1	yes	characteristic, complemented, Sylow
75	C₂²×C₄	—	23	1	no	—
76	C₂²×C₄	—	23	1	no	—
77	C₂²×C₄	—	23	2	no	—
78	C₂²×C₄	C₄×D₂₃	1	2	yes	—
79	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
80	C₂×C₈	—	2	2	no	—
81	C₂×C₈	—	2	2	no	—
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	46	2	no	—
84	C₂×C₈	—	46	2	no	—
85	C₂×C₈	—	46	2	no	—
86	C₄²	—	2	2	no	—
87	C₄²	—	46	2	no	—
88	C₄²	—	46	1	no	—
89	C₄²	—	46	1	no	—
90	C₂³	C₂×C₄×D₂₃	1	1	yes	characteristic
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	46	1	no	—
95	C₂×C₄	—	46	1	no	—
96	C₂×C₄	—	46	2	no	—
97	C₂×C₄	—	46	2	no	—
98	C₂×C₄	—	46	2	no	—
99	C₂×C₄	—	46	2	no	—
100	C₂×C₄	—	46	2	no	—
101	C₂×C₄	#67288	1	2	yes	—
102	C₂×C₄	#67290	1	1	yes	characteristic
103	C₂×C₄	#67290	1	2	yes	—
104	C₂×C₄	#67292	1	1	yes	characteristic
105	C₈	—	4	2	no	—
106	C₈	—	92	2	no	—
107	C₂²	#10625043	1	1	yes	characteristic
108	C₂²	#10625048	1	1	yes	characteristic
109	C₂²	#10625063	1	1	yes	characteristic
110	C₂²	#10625072	1	2	yes	—
111	C₂²	#10625078	1	2	yes	—
112	C₄	—	4	2	no	—
113	C₄	—	92	1	no	—
114	C₄	—	92	2	no	—
115	C₄	—	92	1	no	—
116	C₄	#10625050	1	2	yes	—
117	C₄	#10625069	1	2	yes	—
118	C₂	#13092284	1	1	yes	characteristic
119	C₂	#13092475	1	2	yes	—
120	C₂	#13092476	1	1	yes	characteristic
121	C₂	#13092477	1	2	yes	—
122	C₂	#13092479	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13092263

C₂

yes

maximal, characteristic

#13092172

C₂

yes

maximal, characteristic

#13092171

C₂

yes

maximal

#13092171

C₂

yes

maximal

C₂₃×#336

C₂

yes

maximal, characteristic

C₂²×C₈×C₂₃

C₂²

yes

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂×C₄×Dic₂₃

C₂²

yes

characteristic

C₂×C₄×Dic₂₃

C₂²

yes

characteristic

C₂²×C₂₃⋊C₈

C₂²

yes

—

C₂₃⋊#95

C₄

yes

—

C₂₃⋊#95

C₄

yes

—

C₂²×C₄×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

C₂×C₄

yes

characteristic

C₂²×Dic₂₃

C₂×C₄

yes

characteristic

C₂²×Dic₂₃

C₂×C₄

yes

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₄²×C₂₃

—

C₄×Dic₂₃

—

C₄×Dic₂₃

—

C₄×Dic₂₃

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂³×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

#41

yes

—

C₂×C₄×C₂₃

#41

yes

—

C₂×C₄×C₂₃

yes

C₂×C₄×C₂₃

yes

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₈×C₂₃

—

C₂₃⋊C₈

—

—

C₂²×C₂₃

yes

C₂²×C₂₃

yes

C₂²×C₂₃

yes

C₂²×C₂₃

yes

—

C₂²×C₂₃

#131

yes

—

C₄×C₂₃

—

C₄×C₂₃

#126

yes

—

C₄×C₂₃

#126

yes

—

Dic₂₃

—

Dic₂₃

—

Dic₂₃

—

—

—

—

—

D₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

—

C₂×C₂₃

#433

yes

characteristic

C₂×C₂₃

#433

yes

—

C₂×C₂₃

#433

yes

characteristic

C₂²×C₈

—

C₂²×C₈

C₂×D₂₃

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₂₃

yes

characteristic

#95

—

#95

—

C₂₃

#1840

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₂₃

yes

—

C₂²×C₄

C₂²×D₂₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₂³

C₂×C₄×D₂₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

#67288

yes

—

102

C₂×C₄

#67290

yes

characteristic

103

C₂×C₄

#67290

yes

—

104

C₂×C₄

#67292

yes

characteristic

105

C₈

—

106

C₈

—

107

C₂²

#10625043

yes

characteristic

108

C₂²

#10625048

yes

characteristic

109

C₂²

#10625063

yes

characteristic

110

C₂²

#10625072

yes

—

111

C₂²

#10625078

yes

—

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

#10625050

yes

—

117

C₄

#10625069

yes

—

118

C₂

#13092284

yes

characteristic

119

C₂

#13092475

yes

—

120

C₂

#13092476

yes

characteristic

121

C₂

#13092477

yes

—

122

C₂

#13092479

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₂₃
commutator, socle, prime radical	C₂³×C₂₃	C₂²×C₄
Fitting	C₂₃×#336	C₂
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	#336	D₂₃
hypocenter	C₂₃	#1840
LCS	G ▷ C₂³×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂²×C₄ ◁ #336

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₂₃

commutator, socle, prime radical

C₂³×C₂₃

C₂²×C₄

Fitting

C₂₃×#336

C₂

cosocle

C₂²×C₄×C₂₃

C₂³

D₂₃

C₂₃

G ▷ C₂³×C₂₃ ▷ C₂₃

C₁ ◁ C₂²×C₄ ◁ #336

N	H ≅ G/N	type
74. C₂₃	#1840	semidirect

H ≅ G/N

type

74. C₂₃

#1840

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#13092263	2
#13092172	2
#13092171	2
C₂₃×#336	2
#1840	23

subgroup

index

C₂₃×#336

Subgroups — #15673541

Conjugacy classes of subgroups

Complements

Maximal subgroups